For n[ges ]2, let S(Rn) be the space of Schwartz class functions. The dual space of S(Rn) is denoted by S′(Rn) and is called the tempered distributions. For f∈S(Rn), define fˆ(w)=∫Rne−iw·xf(x)dx and for T∈S′(Rn), define Tˆ∈S′(Rn) by the formula 〈Tˆ, f〉=〈T, fˆ〉 for all f∈S(Rn). For 1[les ]p[les ]∞, let FLp(Rn)= {T∈S′(Rn)[ratio ] Tˆ∈Lp(Rn)}. Let β be a multi-index of nonnegative integers, β=(β1, β2, …, βn−1) with [mid ]β[mid ]= β1+β2+…+βn−1. For a nonnegative integer k and bounded Borel measures σβ on a compact subset K of Rn, [mid ]β[mid ][les ]k, we define T∈S′(Rn) by the formula

formula here

If T is given by (1), then we know that supp(T)⊂K and the order of T is at most k. In particular, when k=0, we say that T is a bounded measure on K. Denote EK(Rn)= {T∈S′(Rn)[ratio ]T is a bounded measure on K} and DK(Rn)= {T∈S′(Rn)[ratio ] supp(T)∪K}. If in addition we assume some connectness property of K, then one can show ([5, th. 2·3·10]) that if T∈DK(Rn), then T does have the representation (1) for some k[ges ]0. The question is when we can say that k=0. From the Hausdorff–Young inequality and the proof of theorem 7·6·6 in [5], one sees that if the interior of K (in Rn) is not empty, then DK(Rn) ∩FLp(Rn) ⊂EK(Rn) if and only if 1[les ]p[les ]2. Much remains to be understood if the interior of K is empty.

In this paper, we seek to prove a similar result for compact subsets of hypersurfaces of Rn, motivated from the differential equations with constant coefficients.